💰 Ab Exponential Regression 計算機

Exponential regression fits a curve of the form y = A · B^{x} to a set of paired observations. It is the right tool whenever a quantity grows or decays by a roughly constant factor for each unit increase in x — population growth, compound interest, radioactive decay, bacterial cultures and many natural processes. This is a universal statistical tool with no region-specific rules.

Enter your data one pair per line as x,y. You need at least two points, the x values must not all be identical, and every y value must be strictly positive (the method takes the natural logarithm of y). Pick the number of display digits, then read off A, B and the correlation coefficient r.

The model is non-linear, but taking logarithms makes it linear: \ln(y) = \ln(A) + x · \ln(B). We therefore run an ordinary least-squares line fit on the transformed points (x_i, \ln y_i). With \bar{x} the mean of x and meanLnY the mean of ln y, define

For the points (1, 2.7), (2, 7.4), (3, 20.1), (4, 54.6), (5, 148.4): n = 5, \bar{x} = 3, meanLnY ≈ 2.99906. S_{xx} = 10, S_{xy} ≈ 10.01167, S_{yy} ≈ 10.02337. So

An exponential regression returns three numbers — A, B and r — that together describe the model y = A · B^{ x}. Here is how to read each one.

The base B: growth or decay

The base B controls the direction and speed of change for every one-unit increase in x:

• B > 1 means growth. Each step in x multiplies y by B, so the curve rises. The per-unit percentage change is (B-1)×100%. For example, B = 1.08 corresponds to 8% growth per unit of x.
• B < 1 means decay. Each step multiplies y by less than one, so the curve falls toward zero. The same formula (B-1)×100% gives a negative result; e.g. B = 0.85 is a -15% change per unit.
• B = 1 is flat. y stays equal to A regardless of x (zero percent change).

The coefficient A: the y-intercept

A is the value of y when x = 0, because A · B^{0} = A. It anchors the curve vertically and represents the starting amount, the initial population, the principal, or the dose at the origin of your x-axis.

The correlation coefficient r: fit on the log scale

This fit works by taking z_i = \ln y_i and running an ordinary linear regression of z against x. Consequently, r measures how well the log-transformed data \ln(y) fall on a straight line — not how well the raw y values fit the curve. A value of r close to +1 or -1 indicates a strong exponential relationship; the sign matches the direction (positive for growth, negative for decay).

Using standard correlation interpretation bands for |r|:

• 0.9 – 1.0: very strong fit — the data closely follow the exponential model.
• 0.7 – 0.9: strong fit — exponential is a good description with some scatter.
• 0.5 – 0.7: moderate fit — a trend exists but other factors are at play.
• below 0.5: weak fit — an exponential model may not be appropriate.

Because r reflects the log-scale fit, a high r does not guarantee small errors on the original y scale; large y values are weighted less after the \ln transform. Always plot the fitted curve against your raw data as a sanity check.